Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T19:16:17.997Z Has data issue: false hasContentIssue false
  • English
  • Français

Totally Integrally Closed Azumaya Algebras

Published online by Cambridge University Press:  20 November 2018

R. Macoosh  and
R. Raphael
R. Macoosh
Affiliation:
Department of Mathematics Concordia University, Loyola Campus 7141 Sherbrooke St. W., Montreal, Quebec H4B 1R6
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Enochs introduced and studied totally integrally closed rings in the class of commutative rings. This article studies the same question for Azumaya algebras, a study made possible by Atterton's notion of integral extensions for non-commutative rings.

The main results are that Azumaya algebras are totally integrally closed precisely when their centres are, and that an Azumaya algebra over a commutative semiprime ring has a tight integral extension that is totally integrally closed. Atterton's integrality differs from that often studied but is very natural in the context of Azumaya algebras. Examples show that the results do not carry over to free normalizing or excellent extensions.

Keywords

16A16
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 4 , 01 December 1990 , pp. 398 - 403
Copyright
Copyright © Canadian Mathematical Society 1990

References

1. Atterton, T. W. Definitions of integral elements and quotient rings over non-commutative rings with identity, J. Austr. Math. Soc, 12, (1972), 433–446.Google Scholar
2. Borho, W. and Weber, V. Zur existenz total ganz abgeschlossener ringerweiterungen, Math. Z., 121, (1971), 55–57.Google Scholar
3. DeMeyer, F. and Ingraham, E. Separable algebras over commutative rings, Springer Verlag, Berlin- Heidelberg-New York, (1971).Google Scholar
4. Enochs, E. Totally integrally closed rings, Proc. Amer, Math. Soc, 19, (1968), 701–706.Google Scholar
5. Macoosh, R. Algebraic closures for commutative rings, master thesis, Concordia University, (1987).Google Scholar
6. Pare, R. and Schelter, W. Finite extensions are integral, Algebra, J., 53, (1978), 477–4479.Google Scholar
7. Parmenter, M. M. and P. Stewart, N. Excellent extensions, Comm. in Algebra (to appear).Google Scholar
8. Procesi, C. Rings with polynomial identities, Dekker, M., New York,(1973).Google Scholar
9. Raphael, R. Algebraic extensions of commutative regular rings, Can. Jour. Math., 22, 6, (1970), 1133– 1155.Google Scholar
10. Zariski, O. and Samuel, P. Commutative algebra, 1, Van Nostrand, Princeton, N. J., (1963).Google Scholar